The magnetized spherical couette system: From numerics to experiments

The study of magnetohydrodynamic (MHD) instabilities occurring in liquid metals, with
imposed differential rotation and magnetic field, is of fundamental importance in the astrophysical
context. MHD instabilities are especially relevant in planets or stars, where electrically conducting
flows are confined within their interiors. Such environments could be modeled by solving the
Navier-Stokes and induction equations with appropriate conditions in a spherical shell composed of
two concentric spheres. In particular, we consider the case where the liquid metal (GaInSn in our
case), bounded by a stationary outer sphere and a uniformly rotating inner sphere, is subjected to an
axial magnetic field. When the aspect ratio of the radii of the two spheres is fixed, only two
parameters, namely, the Reynolds number (associated with the differential rotation) and the
Hartmann number (associated with the applied magnetic field strength), govern the dynamics of the
system (see [1,2] for full details).
For the magnetized spherical Couette system, three different types of instabilities have so far
been identified and characterized by means of numerical simulations (e.g. [1,3]), and also in
experiments (e.g. [2,4]). These instabilities can each be described as a hydrodynamic radial jet
instability, a return flow instability, and a Kelvin-Helmholtz-like Shercliff layer instability. We
provide an overview of these instabilities with a focus on the description and analysis of the
different spatio-temporal symmetries of the MHD flow. In particular, numerical and experimental
bifurcation diagrams of nonlinear waves in the quasi-laminar regime (with moderate differential
rotation) are presented and some numerical tools, related to nonlinear dynamics and chaos theory
[5], are outlined. These tools include the numerical continuation of periodic solutions and their
stability assessment, time series analysis such as the computation of the fundamental frequencies in
one or several spatial dimensions, time dependent frequency spectra, and Poincaré sections.
Our results show how periodic and quasiperiodic MHD flows with two, three and even four
incommensurable frequencies, as well as MHD chaotic flows, are developed following a sequence
of bifurcations from the base state. The knowledge of the different routes to chaos is of fundamental
importance in turbulence theory. In addition, by taking into account the symmetries of the solutions
several regions of multistability (and also hysteretic behavior) are identified in the parameter space
with a good agreement between simulations and experiments, both in their temporal and spatial
structures. Although unstable MHD flows are not experimentally realized, their numerical
computation as in [1,6] provides a more complete picture of the dynamics and aids the
understanding of transient and hysteretic behaviors in experiments.
This work is funded by the European Research Council (ERC), Horizon 2020 research and
innovation programme (grant agreement No. 787544). The authors wish to thank Kevin Bauch for
technical support.

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